Beginning with Edwards' normal form

,

to derive the Hamiltonian form:

.

Each level set

is a quartic elliptic curve with period energy function

.

This Hamiltonian double covers another,

.

The contours of

with

are hyperelliptic curves where the Jacobi elliptic functions naturally arise in the time-parametrization problem. This surface looks astoundingly similar to a model for the plane pendulum,

.

In fact, the second and third Hamiltonians are related by a canonical transformation, which leads to the exact solution for the equations of motion of a plane pendulum.

This can be readily applied, since we need only a first approximation of the Edwards doubly periodic, meromorphic function,

.

One issue remains with this simple form, the calculation of

. As with cubic anharmonic oscillation, the Picard–Fuchs equation is just a hypergeometric differential equation, so that

.

Using this function, we can immediately determine time-dependent solutions of motion along

.

The other two Hamiltonian surfaces have equations of motion that depend on the Jacobi functions

and

,

,

,

,

,

for

and

respectively. The Jacobi functions have the same pole-and-zero structure as

, so it is possible to represent both

and

in terms of

. Restricting the domain to

, the simplest instance suffices,

.

However, the function

is not really designed for this purpose. We need to do some more hacking before this identity works. The function

that satisfies

provides the necessary convergence acceleration. We can then propose the solution

.

Proof that

satisfies the requisite constraint reduces to routine evaluation of a binomial identity [3, 4],

.

Zeilberger's algorithm accepts the right-hand side as input, so the identity can be proven automatically. The RISC Mathematica implementation returns the appropriate hypergeometric parameters [4]. Finally, if

is the parameter of a Jacobi function, the corresponding period ratio for

is

.

Very optimal expressions exist for calculating with the Jacobi functions

. The built-in Mathematica functions

JacobiCN and

JacobiSN can do this. However, in our own timing tests, it appears that evaluation of the Edwards

function is sometimes faster by about a factor of 10.

[3] N. J. A. Sloane.

*The On-Line Encyclopedia of Integer Sequences*. "A002057."

oeis.org/A002057.

[4] N. J. A. Sloane and P. D. Hanna.

*The On-Line Encyclopedia of Integer Sequences*. "A119245."

oeis.org/A119245.

[5] P. Paule and M. Schorn, "A Mathematica Version of Zeilberger’s Algorithm for Proving Binomial Coefficient Identities,"

*Journal of Symbolic Computation*,

**20**(5–6), 1995 pp. 673–698.

doi:10.1006/jsco.1995.1071.